已知f(x）是R上的偶函数,且当x＞=0时,f(x)=2^x-2x^(1/2),又a是函数g(x)=ln(x+1)-2/x的正零点,则f(-2),f(a),f(1.5)的大小关系?

已知f(x）是R上的偶函数,且当x＞=0时,f(x)=2^x-2x^(1/2),又a是函数g(x)=ln(x+1)-2/x的正零点,则f(-2),f(a),f(1.5)的大小关系?
已知f(x）是R上的偶函数,且当x＞=0时,f(x)=2^x-2x^(1/2),又a是函数g(x)=ln(x+1)-2/x的正零点,则f(-2),f(a),f(1.5)的大小关系?

已知f(x）是R上的偶函数,且当x＞=0时,f(x)=2^x-2x^(1/2),又a是函数g(x)=ln(x+1)-2/x的正零点,则f(-2),f(a),f(1.5)的大小关系?
首先f(x)=2^x-2x^(1/2)所以求导f'(x)=2^x*ln2-1/√x
当x大于1是可以很容易看出2^x*ln2是指数增长远远大于1/√x
所以当x大于1是f'(x)大于0所以x大于1是函数单调递增
又因为ln(x+1)和2/x,当x=1.5时ln(x+1)=ln2.5,因为e=2.7所以ln2.5小于1而2/1.5=4/3大于1所以当x=1.5的时候ln(x+1)小于2/x,当x=2时ln3大于1,2/3小于1此时ln(x+1)大于2/x,所以a介于1.5和2之间,又因为当x大于1是f'(x)大于0所以x大于1是函数单调递增,所以f(-2)大于f(a)大于f(1.5)

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